Modular symmetry of localized modes

We study the modular symmetry of localized modes on fixed points of $T^2/\mathbb{Z}_2$ orbifold. First, we find that the localized modes with even (odd) modular weight generally have $\Delta(6n^2)$ ($\Delta'(6n^2)$) modular flavor symmetry. Moreover, when we consider an additional Ansatz, the localized modes with even (odd) modular weight generally enjoy $S_3$ ($S'_4$) modular flavor symmetry, and we show the concrete wave functions of the localized modes.
Comment: 18 pages